Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank–Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.     

Eta-invariants, torsion forms and flat vector bundles

We present a new proof, as well as a C/Q extension (and also certain C/Z extension), of the Riemann–Roch–Grothendieck theorem of Bismut–Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of η-invariants associated to the so-called sub-signature operators. We further show that the Bismut–Lott analytic torsion form can be derived naturally from transgressions of η-forms appearing in the adiabatic limit computations.     

Generalized Bergman kernels on symplectic manifolds

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As a consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut and Lebeau.     

Synthesis,Thermal Properties,and Thermoresponsive Behaviors of Cyclic Poly(2‐(dimethylamino)ethyl Methacrylate)s

This study aims at physicochemical properties of thermo‐ and pH/CO₂‐responsive cyclic homopolymers. Three examples of cyclic poly(2‐(dimethylamino)ethyl methacrylate)s (PDMAs) are synthesized by combining the reversible addition–fragmentation chain transfer process and the Diels–Alder ring‐closure reaction. After cyclization, the glass transition temperature significantly increases (ΔT_g = 51.8–59.7 °C) due to the different configurational entropy and end groups, and the maximum decomposition temperature to lose the pendent groups is drastically decreased from 309 to 278 °C. Effects of polymerization degree, polymer concentration, additive of NaCl, and pH/CO₂ on lower critical solution temperature behaviors of PDMA aqueous solutions are investigated. The cloud points (T_c) of ring PDMAs are usually higher than their linear precursors, and the ΔT_c values obtained under a fixed condition can reach up to 20.7 °C, revealing the crucial role of the topology effect. This study paves the way for unique properties and applications of smart cyclic polymers and their derivatives.

     

The skim of balance theory of 3D garment simulation

The balance theory of 3D garment simulation tries to integrally study the complex system of human body-fabric-environment. The theory includes the geometrical balance, dynamic balance, pressure balance, and heat and moisture transfer balance. In this paper, the ICAD-balance equations are established based on the framework of balance theory and the heat and moisture transfer model of garment. A finite volume method is employed to solve the equations. Experiment results show that ICAD-balance equations can excellently simulate real situation. Therefore, our balance theory provides a good tool for customized clothing design, electronic mirror for clothing-test-platform, and comfortable clothing evaluation research.     

Fluorous solvent for cell culture

Incubation of mouse melanoma B16 cells in fluorous solvents with low boiling point such as perfluoromethylcyclohexane, 1,1,1,3,3,3-hexafluoro-2-propanol, ethylpentafluoropropionate resulted in cell death. However, cells lived up to 2 days in fluorous alcohols such as 2,2,3,3,4,4,5,5-octafluoro-1-pentanol and 3,3,4,4,5,5,6,6,6-nonafluoro-1-hexanol with relatively higher fluorine content. Remarkably, cells survived deprived of nutrition up to 4 days when incubated in 2,2,3,3,4,4,5,5,6,6,6-undecafluoro-1-hexanol or in 2,2,3,3,4,4,5,5,6,6,7,7-dodecafluoroheptanol that have the most number of fluorine atoms (11 and 12, respectively) among the perfluoroalkyl alcohols used, and with boiling points of 128 °C and 169 °C, respectively.     

Determination of the hydrogen-bonding induced local viscosity enhancement in room temperature ionic liquids via femtosecond time-resolved depleted spontaneous emission

The fluorescence depletion dynamics of Rhodamine 700 (R-700) molecules in room temperature ionic liquids (RTILs) 1-ethyl-3-methylimidazolium tetrafluoroborate ([emim][BF(4)]) and 1-hydroxyethyl-3-methylimidazolium tetrafluoroborate ([HOemim][BF(4)]) were investigated to determine the local viscosity of the microenvironment surrounding the fluorescent molecules, which is induced by strong hydrogen bonding interaction between cationic and anionic components in RTILs. The solvation and rotation dynamics of R-700 molecules in RTILs show slower time constants relative to that in conventional protic solvents with the same bulk viscosity, indicating that the probe molecule is facing a more viscous microenvironment in RTILs than in conventional solvents because of the strong hydrogen bonding interaction between cationic and anionic components. In addition, this effect is more pronounced in hydroxyl-functionalized ionic liquid than in the regular RTIL due to the presence of a hydroxyl group as a strong hydrogen bonding donor. The hydrogen-bonding-induced local viscosity enhancement effect related to the heterogeneity character of RTILs is confirmed by the nonexponential rotational relaxation of R-700 determined by time-correlated single photon counting (TCSPC). The geometry of hydrogen bonding complexes with different components and sizes are further optimized by density functional theory methods to show the possible hydrogen-bond networks. A model of the hydrogen-bonding network in RTILs is further proposed to interpret the observed specific solvation and local viscosity enhancement effect in RTILs, where most of the fluoroprobes exist as the free nonbonding species in the RTIL solutions and are surrounded by the hydrogen-bonding network formed by the strong hydrogen-bonding between the cationic and anionic components in RTIL. The optimized geometry of hydrogen bonding complexes with different components and sizes by density functional theory methods confirms the local viscosity enhancement effect deduced from fluorescence depletion and TCSPC experiments. The calculated interaction energies reveal the existence of the stronger hydrogen bonding network in RTILs (especially in hydroxyl-functionalized ionic liquid) than that in conventional protic solvent, which leads to the enhancement effect of local microviscosity, and therefore leads to the slow solvation and rotation dynamics of probe molecules observed in RTILs.     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

NaCl islands decorated with 2D or 3D 3,4,9,10-perylene-tetracarboxylic-dianhydride nanostructures

The formation of PTCDA (3,4,9,10-perylene-tetracarboxylic-dianhydride) nanostrutures on Au(1 1 1)-() covered with NaCl islands has been studied using scanning tunneling microscopy (STM). Atomically resolved STM images show that NaCl grows as (1 0 0)-terminated layers on Au(1 1 1)-(). Local atomic hexagonal packing has also been observed in the NaCl(1 0 0) layer. At submonolayer NaCl coverage, PTCDA forms two-dimensional islands on the Au(1 1 1) surface and nucleate preferentially at the NaCl island step edges. When the Au surface is fully covered with NaCl layers, PTCDA molecules form three-dimensional molecular clusters decorating the step edges of NaCl layers.